Journal of Applied and Computational Mechanics
A General Multibody Approach for the Linear and Nonlinear Stability Analysis of Bicycle Systems. Part I: Methods of Constrained Dynamics
Document Type : Research Paper
Authors
1 Department of Industrial Engineering, University of Salerno, Via Giovanni Paolo II, 132, Fisciano, 84084, Salerno, Italy
2 Spin-Off MEID4 s.r.l., University of Salerno, Via Giovanni Paolo II, 132, Fisciano, 84084, Salerno, Italy
Abstract
This investigation is the first contribution of a two-part research work concerning the theoretical development of a multibody approach to analyze the constrained dynamics of articulated mechanical systems. In this paper, a method for investigating the linear and nonlinear stability of the dynamic behavior of mechanical systems modeled as multibody systems subjected to holonomic and nonholonomic constraints is presented. To this end, the nonlinear equations of motions that assume a complex index-three differential-algebraic form are systematically formulated and directly linearized by using an automatic procedure based on a hybrid symbolic-numeric approach devised in this work. The proposed stability analysis method, therefore, is based on the formulation of a generalized eigenvalue problem and represents a viable computer-aided approach suitable for analyzing multibody mechanical systems having different degrees of complexity. Furthermore, an extension of the generalized coordinate partitioning algorithm is introduced in this paper for handling nonholonomic multibody systems leading to a robust and general multibody computational procedure referred to as the Robust Generalized Coordinate Partitioning Algorithm (RGCPA). Since the methodologies employed in this paper to study the stability of multibody mechanical systems are general and versatile, they can be easily implemented in general-purpose multibody computer programs and readily used to analyze several mechanical applications having engineering interest.
Keywords
- Multibody System Dynamics
- Holonomic and Nonholonomic Constraints
- Robust Generalized Coordinate Partitioning Algorithm
- Stability Analysis
Main Subjects
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[9] Pappalardo, C. M., Guida, D., On the Dynamics and Control of Underactuated Nonholonomic Mechanical Systems and Applications to Mobile Robots, Archive of Applied Mechanics, 89(4), 2019, 669-698.
[10] Vlasenko, D., Kasper, R., Generation of equations of motion in reference frame formulation for fem models, Engineering Letters, 16(4), 2008, 537-544.
[11] Ruggiu, M., Kinematic analysis of a fully decoupled translational parallel manipulator, Robotica, 27(07), 2009, 961.
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[18] Mariti, L., Belfiore, N.P., Pennestrí, E., Valentini, P.P., Comparison of solution strategies for multibody dynamics equations, International Journal for Numerical Methods in Engineering, 88(7), 2011, 637-656.
[19] Barbagallo, R., Sequenzia, G., Cammarata, A., Oliveri, S.M., Fatuzzo, G., Redesign and multibody simulation of a motorcycle rear suspension with eccentric mechanism, International Journal on Interactive Design and Manufacturing (IJIDeM), 12(2), 2018, 517-524.
[20] Palomba, I., Richiedei, D., Trevisani, A., Two-stage approach to state and force estimation in rigid-link multibody systems, Multibody System Dynamics, 39(1-2), 2017, 115-134.
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[22] Tasora, A., Mangoni, D., Negrut, D., Serban, R., Jayakumar, P., Deformable soil with adaptive level of detail for tracked and wheeled vehicles, International Journal of Vehicle Performance, 5(1), 2019, 60-76.
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[28] Moore, J.K., Human control of a bicycle, Davis, CA, University of California, Davis, 2012.
[29] Franke, G., Suhr, W., and Rieß, F., An advanced model of bicycle dynamics, European Journal of Physics, 11(2), 1990, 116.
[30] Leine, R.I., The historical development of classical stability concepts: Lagrange, Poisson and Lyapunov stability, Nonlinear Dynamics, 59(1-2), 2010, 173-182.
[31] Vukic, Z., Kuljaca, L.,Donlagic, D., Tesnjak, S., Nonlinear Control Systems, Marcel Dekker, Inc., 2003.
[32] Jain, A., Multibody Graph Transformations and Analysis: Part I: Tree Topology Systems, Nonlinear dynamics, 67(4), 2012, 2779-2797.
[33] Jain, A., Multibody Graph Transformations and Analysis: Part II: Closed-Chain Constraint Embedding, Nonlinear dynamics, 67(4), 2012, 2153-2170.
[34] Ripepi, M., Masarati, P., Reduced order models using generalized eigenanalysis, Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 225(1), 2011, 52-65.
[35] Lehner, M. Eberhard, P., A two-step approach for model reduction in flexible multibody dynamics, Multibody System Dynamics, 17(2-3), 2007, 157-176.
[36] Koutsovasilis, P. Beitelschmidt, M., Comparison of model reduction techniques for large mechanical systems: A study on an elastic rod, Multibody System Dynamics, 20(2), 2008, 111-128.
[37] Maddio, P.D., Meschini, A., Sinatra, R., Cammarata, A., An optimized form-finding method of an asymmetric large deployable reflector, Engineering Structures, 181, 2019, 27-34.
[38] Nikravesh, P.E. Gim, G., Ride and stability analysis of a sports car using multibody dynamic simulation, Mathematical and Computer Modeling, 14, 1990, 953-958.
[39] Kim, J.K., Han, J.H., A multibody approach for 6-DOF flight dynamics and stability analysis of the hawkmoth Manduca sexta, Bioinspiration and Biomimetics., 9(1), 2014, 016011.
[40] Sun, M., Wang, J., Xiong, Y., Dynamic flight stability of hovering insects, Acta Mechanica Sinica, 23(3), 2007, 231-246.
[41] Bauchau, O.A., Wang, J., Stability analysis of complex multibody systems, Journal of Computational and Nonlinear Dynamics, 1, 2006, 71-80.
[42] Cuadrado, J., Vilela, D., Iglesias, I., Martín, A. and Peña, A., A multibody model to assess the effect of automotive motor in-wheel configuration on vehicle stability and comfort, Proc ECCOMAS Themat Conf Multibody Dyn, 2013, 1083-1092.
[43] Escalona, J.L., Chamorro, R., Stability analysis of vehicles on circular motions using multibody dynamics, Nonlinear Dynamics, 53(3), 2008, 237-250.
[44] Quaranta, G., Mantegazza, P., Masarati, P., Assessing the local stability of periodic motions for large multibody non-linear systems using proper orthogonal decomposition, Journal of Sound and Vibration, 271(3-5), 2004, 1015-1038.
[45] Masarati, P., Direct eigenanalysis of constrained system dynamics, Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 223(4), 2010, 335-342.
[46] Negrut, D., Ortiz, J.L., A practical approach for the linearization of the constrained multibody dynamics equations, Journal of Computational and Nonlinear Dynamics, 1(3), 2006, 230-239.
[47] Ortiz, J.L., Negrut, D., Exact linearization of multibody systems using user-defined coordinates, SAE Transactions, 2006, 501-509.
[48] Nichkawde, C., Harish, P.M., Ananthkrishnan, N., Stability analysis of a multibody system model for coupled slosh-vehicle dynamics, Journal of Sound and Vibration, 275(3-5), 2004, 1069-1083.
[49] Bencsik, L., Kovács, L.L., Zelei, A., Stabilization of Internal Dynamics of Underactuated Systems by Periodic Servo-Constraints, International Journal of Structural Stability and Dynamics, 17(05), 2017, 1740004.
[50] Haug, E.J., Computer aided kinematics and dynamics of mechanical systems, Allyn and Bacon, Boston, 1989.
[51] Pappalardo, C.M., A natural absolute coordinate formulation for the kinematic and dynamic analysis of rigid multibody systems, Nonlinear Dynamics, 81(4), 2015, 1841-1869.
[52] Wehage, R.A., Haug, E.J., Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems, Journal of Mechanical Design, 104(1), 1982, 247-255.
[53] Nikravesh, P.E., Some methods for dynamic analysis of constrained mechanical systems: a survey, Computer aided analysis and optimization of mechanical system dynamics, 1984, 351-368.
[54] Nikravesh, P.E., Haug, E.J., Generalized coordinate partitioning for analysis of mechanical systems with nonholonomic constraints, Journal of Mechanical Design, Transactions Of the ASME, 105(3), 1983, 379-384.
[55] Wehage, K.T., Wehage, R.A., Ravani, B., Generalized coordinate partitioning for complex mechanisms based on kinematic substructuring, Mechanism and Machine Theory, 92, 2015, 464-483.
[56] Nada, A.A., Bashiri, A.H., Selective Generalized Coordinates Partitioning Method for Multibody Systems With Non-Holonomic Constraints, Proceedings of the ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, August 2017, American Society of Mechanical Engineers Digital Collection, 2017.
[57] Marques, F., Souto, A.P., Flores, P., On the constraints violation in forward dynamics of multibody systems, Multibody System Dynamics, 39(4), 2017, 385-419.
[58] Masarati, P., Constraint stabilization of mechanical systems in ordinary differential equations form, Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 225(1), 2011, 12-33.
[59] Terze, Z., and Naudet, J., Structure of optimized generalized coordinates partitioned vectors for holonomic and non-holonomic systems, Multibody system dynamics, 24(2), 2010, 203-218.
[60] Zenkov, D.V., Bloch, A.M., Marsden, J.E., The energy-momentum method for the stability of non-holonomic systems, Dynamics and Stability of Systems, 13(2), 1998, 123-165.
[61] Ruina, A., Nonholonomic stability aspects of piecewise holonomic systems, Reports on mathematical physics, 42(1-2), 1998, 91-100.
[62] Borisov, A.V., Mamayev, I.S., The dynamics of a Chaplygin sleigh, Journal of Applied Mathematics and Mechanics, 73(2), 2009, 156-161.
[63] Pappalardo, C.M., Lettieri, A., Guida, D., Stability analysis of rigid multibody mechanical systems with holonomic and nonholonomic constraints, Archive of Applied Mechanics, 90, 2020, 1961-2005.
[64] Pappalardo, C.M., Guida, D., A comparative study of the principal methods for the analytical formulation and the numerical solution of the equations of motion of rigid multibody systems, Archive of Applied Mechanics, 88(12), 2018, 2153-2177.
[65] Pappalardo, C.M., De Simone, M.C., Guida, D., Multibody Modeling and Nonlinear Control of the Pantograph/Catenary System, Archive of Applied Mechanics, 89(8), 2019, 1589-1626.
[66] Rivera, Z.B., De Simone, M.C., Guida, D., Unmanned Ground Vehicle Modelling in Gazebo/ROS-Based Environments, Machines, 7(2), 2019, 42.
[67] De Simone, M., Rivera, Z., Guida, D., Obstacle Avoidance System for Unmanned Ground Vehicles by using Ultrasonic Sensors, Machines, 6(2), 2018, 18.
[68] De Simone, M.C., Guida, D., Control Design for an Under-Actuated UAV Model, FME Transactions, 46(4), 2018, 443-452.
[69] De Simone, M.C., Guida, D., Identification and Control of a Unmanned Ground Vehicle by using Arduino, UPB Scientific Bulletin, Series D: Mechanical Engineering, 80(1), 2018, 141-154.
[70] De Simone, M.C., Rivera, Z.B., Guida, D., Finite Element Analysis on Squeal-Noise in Railway Applications, FME Transactions, 46(1), 2018, 93-100.
[2] Blundell, M., Harty, D., Multibody systems approach to vehicle dynamics, Elsevier, Oxford, UK, 2004.
[3] Shabana, A.A., Computational dynamics, John Wiley & Sons, London, UK, 2009.
[4] Nikravesh, P.E., Computer-aided analysis of mechanical systems, Prentice-Hall, Inc., New Jersey, USA, 1988.
[5] Cajic, M. S., Lazarevic, M. P., Determination of joint reactions in a rigid multibody system, two different approaches, FME Transactions, 44(2), 2016, 165-173.
[6] Meli, E., Magheri, S., Malvezzi, M., Development and implementation of a differential elastic wheel–rail contact model for multibody applications, Vehicle system dynamics, 49(6), 2011, 969-1001.
[7] Ruggiu, M., Kong, X., Reconfiguration Analysis of a 3-DOF Parallel Mechanism, Robotics, 8(3), 2019, 66.
[8] Kong, X., Reconfiguration analysis of a 3-DOF parallel mechanism using Euler parameter quaternions and algebraic geometry method, Mechanism and Machine Theory, 74, 2014, 188-201.
[9] Pappalardo, C. M., Guida, D., On the Dynamics and Control of Underactuated Nonholonomic Mechanical Systems and Applications to Mobile Robots, Archive of Applied Mechanics, 89(4), 2019, 669-698.
[10] Vlasenko, D., Kasper, R., Generation of equations of motion in reference frame formulation for fem models, Engineering Letters, 16(4), 2008, 537-544.
[11] Ruggiu, M., Kinematic analysis of a fully decoupled translational parallel manipulator, Robotica, 27(07), 2009, 961.
[12] Geier, A., Kebbach, M., Soodmand, E., Woernle, C., Kluess, D., Bader, R., Neuro-musculoskeletal flexible multibody simulation yields a framework for efficient bone failure risk assessment, Scientific reports, 9(1), 2019, 1-15.
[13] Kebbach, M., Grawe, R., Geier, A., Winter, E., Bergschmidt, P., Kluess, D., D’Lima, D., Woernle, C., Bader, R., Effect of surgical parameters on the biomechanical behaviour of bicondylar total knee endoprostheses - A robot-assisted test method based on a musculoskeletal model, Scientific Reports, 9(1), 2019, 1-11.
[14] Nitschke, M., Dorschky, E., Heinrich, D., Schlarb, H., Eskofier, B.M., Koelewijn, A.D., Van Den Bogert, A.J., Efficient trajectory optimization for curved running using a 3D musculoskeletal model with implicit dynamics, Scientific Reports, 10(1), 2020, 1-12.
[15] Eberhard, P., Schiehlen, W., Computational dynamics of multibody systems: history, formalisms, and applications, Journal of computational and nonlinear dynamics, 1(1), 2006, 3-12.
[16] Meijaard, J.P., Papadopoulos, J.M., Ruina, A., Schwab, A.L., Linearized dynamics equations for the balance and steer of a bicycle: A benchmark and review, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 463(2084), 2007, 1955–1982.
[17] Kooijman, J.D.G., Schwab, A.L., Meijaard, J.P., Experimental validation of a model of an uncontrolled bicycle, Multibody System Dynamics, 19(1-2), 2008, 115-132.
[18] Mariti, L., Belfiore, N.P., Pennestrí, E., Valentini, P.P., Comparison of solution strategies for multibody dynamics equations, International Journal for Numerical Methods in Engineering, 88(7), 2011, 637-656.
[19] Barbagallo, R., Sequenzia, G., Cammarata, A., Oliveri, S.M., Fatuzzo, G., Redesign and multibody simulation of a motorcycle rear suspension with eccentric mechanism, International Journal on Interactive Design and Manufacturing (IJIDeM), 12(2), 2018, 517-524.
[20] Palomba, I., Richiedei, D., Trevisani, A., Two-stage approach to state and force estimation in rigid-link multibody systems, Multibody System Dynamics, 39(1-2), 2017, 115-134.
[21] Bruni, S., Meijaard, J.P., Rill, G., Schwab, A.L., State-of-the-art and challenges of railway and road vehicle dynamics with multibody dynamics approaches, Multibody System Dynamics, 49, 2020, 1-32.
[22] Tasora, A., Mangoni, D., Negrut, D., Serban, R., Jayakumar, P., Deformable soil with adaptive level of detail for tracked and wheeled vehicles, International Journal of Vehicle Performance, 5(1), 2019, 60-76.
[23] Popp, K., Schiehlen, W.O., Ground Vehicle Dynamics, Springer, Berlin, 2010.
[24] Whipple, F.J., The stability of the motion of a bicycle, Quarterly Journal of Pure and Applied Mathematics, 30(120), 1899, 312-348.
[25] Carvallo, M.E., Theorie de mouvement du monocycle et de la bycyclette, MPublishing, 1901.
[26] Cossalter, V., Motorcycle dynamics, Second Edition, Lulu. Com, 2006.
[27] Moore, J., Hubbard, M., Parametric study of bicycle stability (P207), The engineering of sport, 7, 2008, 311-318.
[28] Moore, J.K., Human control of a bicycle, Davis, CA, University of California, Davis, 2012.
[29] Franke, G., Suhr, W., and Rieß, F., An advanced model of bicycle dynamics, European Journal of Physics, 11(2), 1990, 116.
[30] Leine, R.I., The historical development of classical stability concepts: Lagrange, Poisson and Lyapunov stability, Nonlinear Dynamics, 59(1-2), 2010, 173-182.
[31] Vukic, Z., Kuljaca, L.,Donlagic, D., Tesnjak, S., Nonlinear Control Systems, Marcel Dekker, Inc., 2003.
[32] Jain, A., Multibody Graph Transformations and Analysis: Part I: Tree Topology Systems, Nonlinear dynamics, 67(4), 2012, 2779-2797.
[33] Jain, A., Multibody Graph Transformations and Analysis: Part II: Closed-Chain Constraint Embedding, Nonlinear dynamics, 67(4), 2012, 2153-2170.
[34] Ripepi, M., Masarati, P., Reduced order models using generalized eigenanalysis, Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 225(1), 2011, 52-65.
[35] Lehner, M. Eberhard, P., A two-step approach for model reduction in flexible multibody dynamics, Multibody System Dynamics, 17(2-3), 2007, 157-176.
[36] Koutsovasilis, P. Beitelschmidt, M., Comparison of model reduction techniques for large mechanical systems: A study on an elastic rod, Multibody System Dynamics, 20(2), 2008, 111-128.
[37] Maddio, P.D., Meschini, A., Sinatra, R., Cammarata, A., An optimized form-finding method of an asymmetric large deployable reflector, Engineering Structures, 181, 2019, 27-34.
[38] Nikravesh, P.E. Gim, G., Ride and stability analysis of a sports car using multibody dynamic simulation, Mathematical and Computer Modeling, 14, 1990, 953-958.
[39] Kim, J.K., Han, J.H., A multibody approach for 6-DOF flight dynamics and stability analysis of the hawkmoth Manduca sexta, Bioinspiration and Biomimetics., 9(1), 2014, 016011.
[40] Sun, M., Wang, J., Xiong, Y., Dynamic flight stability of hovering insects, Acta Mechanica Sinica, 23(3), 2007, 231-246.
[41] Bauchau, O.A., Wang, J., Stability analysis of complex multibody systems, Journal of Computational and Nonlinear Dynamics, 1, 2006, 71-80.
[42] Cuadrado, J., Vilela, D., Iglesias, I., Martín, A. and Peña, A., A multibody model to assess the effect of automotive motor in-wheel configuration on vehicle stability and comfort, Proc ECCOMAS Themat Conf Multibody Dyn, 2013, 1083-1092.
[43] Escalona, J.L., Chamorro, R., Stability analysis of vehicles on circular motions using multibody dynamics, Nonlinear Dynamics, 53(3), 2008, 237-250.
[44] Quaranta, G., Mantegazza, P., Masarati, P., Assessing the local stability of periodic motions for large multibody non-linear systems using proper orthogonal decomposition, Journal of Sound and Vibration, 271(3-5), 2004, 1015-1038.
[45] Masarati, P., Direct eigenanalysis of constrained system dynamics, Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 223(4), 2010, 335-342.
[46] Negrut, D., Ortiz, J.L., A practical approach for the linearization of the constrained multibody dynamics equations, Journal of Computational and Nonlinear Dynamics, 1(3), 2006, 230-239.
[47] Ortiz, J.L., Negrut, D., Exact linearization of multibody systems using user-defined coordinates, SAE Transactions, 2006, 501-509.
[48] Nichkawde, C., Harish, P.M., Ananthkrishnan, N., Stability analysis of a multibody system model for coupled slosh-vehicle dynamics, Journal of Sound and Vibration, 275(3-5), 2004, 1069-1083.
[49] Bencsik, L., Kovács, L.L., Zelei, A., Stabilization of Internal Dynamics of Underactuated Systems by Periodic Servo-Constraints, International Journal of Structural Stability and Dynamics, 17(05), 2017, 1740004.
[50] Haug, E.J., Computer aided kinematics and dynamics of mechanical systems, Allyn and Bacon, Boston, 1989.
[51] Pappalardo, C.M., A natural absolute coordinate formulation for the kinematic and dynamic analysis of rigid multibody systems, Nonlinear Dynamics, 81(4), 2015, 1841-1869.
[52] Wehage, R.A., Haug, E.J., Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems, Journal of Mechanical Design, 104(1), 1982, 247-255.
[53] Nikravesh, P.E., Some methods for dynamic analysis of constrained mechanical systems: a survey, Computer aided analysis and optimization of mechanical system dynamics, 1984, 351-368.
[54] Nikravesh, P.E., Haug, E.J., Generalized coordinate partitioning for analysis of mechanical systems with nonholonomic constraints, Journal of Mechanical Design, Transactions Of the ASME, 105(3), 1983, 379-384.
[55] Wehage, K.T., Wehage, R.A., Ravani, B., Generalized coordinate partitioning for complex mechanisms based on kinematic substructuring, Mechanism and Machine Theory, 92, 2015, 464-483.
[56] Nada, A.A., Bashiri, A.H., Selective Generalized Coordinates Partitioning Method for Multibody Systems With Non-Holonomic Constraints, Proceedings of the ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, August 2017, American Society of Mechanical Engineers Digital Collection, 2017.
[57] Marques, F., Souto, A.P., Flores, P., On the constraints violation in forward dynamics of multibody systems, Multibody System Dynamics, 39(4), 2017, 385-419.
[58] Masarati, P., Constraint stabilization of mechanical systems in ordinary differential equations form, Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 225(1), 2011, 12-33.
[59] Terze, Z., and Naudet, J., Structure of optimized generalized coordinates partitioned vectors for holonomic and non-holonomic systems, Multibody system dynamics, 24(2), 2010, 203-218.
[60] Zenkov, D.V., Bloch, A.M., Marsden, J.E., The energy-momentum method for the stability of non-holonomic systems, Dynamics and Stability of Systems, 13(2), 1998, 123-165.
[61] Ruina, A., Nonholonomic stability aspects of piecewise holonomic systems, Reports on mathematical physics, 42(1-2), 1998, 91-100.
[62] Borisov, A.V., Mamayev, I.S., The dynamics of a Chaplygin sleigh, Journal of Applied Mathematics and Mechanics, 73(2), 2009, 156-161.
[63] Pappalardo, C.M., Lettieri, A., Guida, D., Stability analysis of rigid multibody mechanical systems with holonomic and nonholonomic constraints, Archive of Applied Mechanics, 90, 2020, 1961-2005.
[64] Pappalardo, C.M., Guida, D., A comparative study of the principal methods for the analytical formulation and the numerical solution of the equations of motion of rigid multibody systems, Archive of Applied Mechanics, 88(12), 2018, 2153-2177.
[65] Pappalardo, C.M., De Simone, M.C., Guida, D., Multibody Modeling and Nonlinear Control of the Pantograph/Catenary System, Archive of Applied Mechanics, 89(8), 2019, 1589-1626.
[66] Rivera, Z.B., De Simone, M.C., Guida, D., Unmanned Ground Vehicle Modelling in Gazebo/ROS-Based Environments, Machines, 7(2), 2019, 42.
[67] De Simone, M., Rivera, Z., Guida, D., Obstacle Avoidance System for Unmanned Ground Vehicles by using Ultrasonic Sensors, Machines, 6(2), 2018, 18.
[68] De Simone, M.C., Guida, D., Control Design for an Under-Actuated UAV Model, FME Transactions, 46(4), 2018, 443-452.
[69] De Simone, M.C., Guida, D., Identification and Control of a Unmanned Ground Vehicle by using Arduino, UPB Scientific Bulletin, Series D: Mechanical Engineering, 80(1), 2018, 141-154.
[70] De Simone, M.C., Rivera, Z.B., Guida, D., Finite Element Analysis on Squeal-Noise in Railway Applications, FME Transactions, 46(1), 2018, 93-100.
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